Drawing on the work of Gattegno, it is suggested that a powerful way of teaching mathematics is to introduce symbols as relationships between visible or tangible resources. The symbols are abstract (formal) from the beginning and yet there are concrete resources to support their use. Drawing on data from a research project in primary schools in…

This guidance report focuses on the teaching of mathematics to pupils in Key Stages 2 and 3. It is not intended to provide a comprehensive guide to mathematics teaching. We have made recommendations where there are research findings that schools can use to make a significant difference to pupils' learning, and have focused on the questions that…

The National Mathematics Advisory Panel specifies that children should be able to fluently add and subtract whole numbers by the end of third grade. Mathematical understanding of addition and subtraction emerges well before formal mathematic instruction begins, and when it emerges it reflects competence in a number of prerequisite skills and…

Analyzing student work can help teachers improve the teaching and learning of mathematics. Fortunately, the four authors have experienced professional development that supports teachers' professional growth and provides tools to begin the challenge of analyzing student work. However, as they began to assume leadership roles in their schools and…

With the introduction of arithmetic as a compulsory part of the elementary school curriculum in late eighteenth- and early nineteenth-century Germany, textbooks and pedagogical methods were wanted. Relying on the traditions of the Rechenbucher and informed by the demand for method found in Wolffian style advanced textbooks, the first generation of…

Describes the work that students and teachers do to develop computational fluency for subtraction. Examines the orchestration of whole-class discourse and presents a collection of common strategies. (Author/NB)

Suggests ways to increase students' computational fluency using concrete materials, engaging tasks, and reflection time to increase number automaticity, flexibility in thinking about numbers, and use of efficient problem-solving strategies to find sums and differences. (Author/NB)

This classroom scholarship report presents a group of elementary students' experiences learning the traditional long division algorithm. The traditional long division algorithm is often taught mechanically, resulting in the student's performance of step-by-step procedures with no or weak understanding of the concept. While noting some initial…

This book introduces case studies in mathematics education that provide teachers with opportunities to look at someone else's teaching in an objective way and analyze their own teaching with a critical eye. In addition, this book addresses many of the topics considered the hardest to teach and learn, including place value, number sentences, basic…

Provides an alternative to traditional instruction in multiplication and division to develop computational fluency in students. (Author/NB)

Six easy-to-learn strategies are offered to help students with learning problems in mathematics make error-free subtraction calculations when either the subtrahend or remainder is a 9, 8, or 7 and when the minuend is in the teens. (CB)

A fraction table consists of a large number of lines taped to a table. Strips of paper that represent fractions can be manipulated on it to solve problems. Uses of the table for discovering fractions are discussed, considering equivalent fractions and adding, subtracting, multiplying, and dividing fractions. (JN)

Suggests that strategy choice is a misleading characterization of efficient mental calculation and that teaching mental calculation methods as a whole is not conducive to flexibility. Proposes an alternative in which calculation is thought of as an interaction between noticing and knowledge. Presents an associated teaching approach to promote…

Presents the listing (and documentation) for an arithmetic drill program with color graphics and sound. The program offers simple arithmetic problems illustrated by colorful dinosaurs. A buzzer sounds if an answer is incorrect while a short piece of dinosaur music is heard if an answer is correct. (JN)

This paper provides some guidance as to what to listen for to help students make sense of expressions in ways that connect to their ideas and honestly address the mathematics of rational numbers. It offers a reasonable initial answer to the question, "Where do students' ideas about fractions and ratios come from, and how can we work productively…